Lab V
Simple Harmonic Motion

Continued

II. (a) Use Maple to solve the differential
equation describing a damped harmonic oscillator with parameters the same as in
I(a) except now there is a damping force characterized by -bv(t) where the damping constant,
β= b/2m.

Remember in part Ia there was a mass, m, attached to a spring of force
constant, k. The initial position of the mass was at x0 and its initial
velocity v0 and the angular frequency was w0 = (k/m)^{1}^{/2}.

**Hint:**You will
solve the equation diff(x(t),t,t) +
2*beta*diff(x(t),t) + omega0^2*x(t) = 0 with x(0)=x0,D(x)(0)=v0 (like last
week's lab);

You can also solve the DFQ using Solve DE Intercatively. Go
ahead and do this too (just for fun).

Assign your solution

(b) Let x0 = 1, v0 = 0, and w_{0} = 1.
Plot the position x(t) for t from 0 to 4p for the case of β = 0.25 (underdamping),
β = 1 (critical damping), β = 4 (overdamping).

**Hints:** In the case of beta = 1, you will need to plot the limit as beta goes
to 1, eg
plot(Limit(x(t),beta = 1),t = 0..4*Pi);.
Note that this will take Maple some time to do.
Alternatively, you can just let beta = 0.99999
You might use X:= unapply(x(t), t,beta); and then plot(X(t,0.25), t = 0..4*Pi);

You probably should restart before part c

(c) For the underdamping
case of part (b), obtain a simple expression for x(t)
in terms of cosines, sines and e^{-}^{b}^{t} by
defining w_{1}^{2} = w_{0}^{2} - β^{
2}. Obtain the simple expression for the case
when omega1 = 0.9375, beta = 0.25, x0=1.0, v0=0.0

**Hints**
Solve the same equation for x(t) in terms of omega0 exactly as you did and then assign
the solution and assume omega1, beta are > 0. Then use
subs(omega0=sqrt(beta^2+omega1^2),x(t)):
x2(t) :=simplify(evalc(subs({omega1 = 0.9375, beta = 0.25,
x0=1.0, v0=0.0},%))); If asked, choose table. You may have to use x2 and then unapply to get x2(t).

(d) Make a phase plot for this showing the
decay in the position and velocity.

**Hint:** Define X(t) and V(t): X:= unapply(x2(t),t);
V:= unapply(diff(x2(t),t),t); and plot them like last
time